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arXiv:cond-mat/0207157 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Percolation in a random environment

Róbert Juhász, Ferenc Iglói

Published Nov 19, 2002• 9 citations
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Abstract

We consider bond percolation on the square lattice with perfectly correlated random probabilities. According to scaling considerations, mapping to a random walk problem and the results of Monte Carlo simulations the critical behavior of the system with varying degree of disorder is governed by new, random fixed points with anisotropic scaling properties. For weaker disorder both the magnetization and the anisotropy exponents are nonuniversal, whereas for strong enough disorder the system scales into an infinite randomness fixed point in which the critical exponents are exactly known.

DOI
10.1103/physreve.66.056113

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We consider bond percolation on the square lattice with perfectly correlated random probabilities. According to scaling considerations, mapping to a random walk problem and the results of Monte Carlo simulations the critical behavior of the system with varying degree of disorder is governed by new, random fixed points with anisotropic scaling properties. For weaker disorder both the magnetization and the anisotropy exponents are nonuniversal, whereas for strong enough disorder the system scales into an infinite randomness fixed point in which the critical exponents are exactly known.
In complete analogy with thermal critical phenomena, it is expected that anisotropic percolation is in the same universality class as the isotropic one. However the previous result, obtained from the closed form conventional cell real space renormalization group method on the square lattice, the isotropic fixed point is completely unstable is known. We examine this in the large cell limit by using a Monte Carlo renormalization group method and show that the scaling exponent associated with anisotropy is not relevant.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.
Scale-invariant universal crossing probabilities are studied for critical anisotropic systems in two dimensions. For weakly anisotropic standard percolation in a rectangular-shaped system, Cardy's exact formula is generalized using a length-rescaling procedure. For strongly anisotropic systems in 1+1 dimensions, exact results are obtained for the random walk with absorbing boundary conditions, which can be considered as a linearized mean-field approximation for directed percolation. The bond and site directed percolation problem is itself studied numerically via Monte Carlo simulations on the diagonal square lattice with either free or periodic boundary conditions. A scale-invariant critical crossing probability is still obtained, which is a universal function of the effective aspect ratio r eff = c r , where r = L / t z , z is the dynamical exponent and c is a non-universal amplitude.
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at Tc{\mathit{T}}_{\mathit{c}}Tc​, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.
Percolation with bonds oriented along an external field direction is considered. Assuming that the second-order type phase transition picture can be applied in the vicinity of the critical point the threshold probability and the correlation length exponent nu 0 is calculated on the square lattice by Monte Carlo techniques using finite-size scaling arguments. It is found that nu 0 =1.65+or-0.06, i.e. the orientated percolation belongs to a different university class from the unorientated one. The critical probabilities for several two-dimensional lattices are determined.